Arcs, Valuations, and Multiplier Ideals on Algebraic Varieties

University Of Utah, Salt Lake City UT

Investigators

Abstract

This research project is in the field of algebraic geometry. At the core of algebraic geometry is the quest for classification of algebraic varieties, which are sets of solutions of systems of polynomial equations (or various generalizations of this concept). Classification roughly splits into two parts: birational classification, in which varieties that look alike almost everywhere are organized in the same class, and moduli theory, where continuous deformations of varieties are studied. The minimal model program is the major tool for the birational classification of algebraic varieties, and it has important applications to the theory of moduli. Designed over the model of classification of surfaces, the minimal model program has become one of the major trends in algebraic geometry, earning S. Mori the Fields Medal for work on three dimensional varieties. Recently, there has been groundbreaking progress which brings the subject very close to a complete program in all dimensions. There are however many fundamental questions that still remain unanswered, and some of these constitute the main objectives of this research project. The local study of algebraic varieties and their singularities is a natural component of this study. The project has two main objectives. The first objective concerns a conjecture of Shokurov from 1988 which predicts that a certain local invariant of singularities, known as the minimal log discrepancy, varies semicontinuously. This property is closely connected to a conjecture on termination of flips which has implications to the minimal model program, and relates to a series of specific questions concerning the geometry of spaces of arcs and their connections to valuation spaces and Berkovich analytic spaces. The second objective deals with multiplier ideals on algebraic varieties. There are various notions of multiplier ideals on singular varieties, and the PI will explore a unifying approach which incorporates all theories and explains their connections. This part of the project contains various applications, from a Reider-type theorem on the global generation of twists of the dualizing sheaf of a normal surface, to the comparisons between certain notions of singularities defined in characteristic zero and in positive characteristics. The project also includes educational activities and other broader impacts. These include the participation in outreach programs, various forms of dissemination including the writing of a book on the birational geometry of algebraic varieties, and the organization of several activities at various levels, ranging from undergraduate summer schools to international conferences.